Enough to get by! I worked on this throughout university, and now at my startup. I was slightly blasé however! The corpus is tiny (173 episodes: http://www.chakoteya.net/ds9/episodes.htm), so a topic model is unlikely to yield anything valuable. There are probably around 10-15 arcs, and simple clustering could be better -- but this is purely hypothetical. In this case, it's simply curiosity.

If you're interested in tools, Mallet (http://mallet.cs.umass.edu/) is a fairly good place to start, and the original LDA paper by Blei, Ng & Jordan (http://machinelearning.wustl.edu/mlpapers/paper_files/BleiNJ...) is a great academic starting point.